Question

19 a landscaper is putting a border on each side of a triangular garden. she plots the vertices of the garden (-2, 3), (5, 5) and (7, -3) on a coordinate grid. she estimates that she will need 2 feet of material for the border for every unit of length represented in the graph of the garden. (a) part a which expression can be used to determine the amount of material needed for the border on the side of the garden represented by the vertices (5, 5) and (7, -3) on the graph? a 2\sqrt{(5 - 7)^2+(5 - (-3))^2} b 2\sqrt{(\frac{5 - 7}{2})^2+(\frac{5 - (-3)}{2})^2} c 2\sqrt{(5 + 7)^2+(5 + (-3))^2} d 2\sqrt{(\frac{5 + 7}{2})^2+(\frac{5 + (-3)}{2})^2} (b) part b the material for the border costs $1.50 per foot. to the nearest dollar, what is the total cost of the material needed to put a border around the entire garden? a $40 b $52 c $79

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$. Here, $x_1 = 5,y_1 = 5,x_2=7,y_2=- 3$. And she needs 2 feet of material for every unit of length, so the length of material for the side with endpoints $(5,5)$ and $(7,-3)$ is $2\sqrt{(5 - 7)^2+(5-(-3))^2}$. So the answer for part A is A.

Step2: Find lengths of all sides

Let $A(-2,3)$, $B(5,5)$, $C(7,-3)$.
Length of $AB$: $d_{AB}=2\sqrt{(-2 - 5)^2+(3 - 5)^2}=2\sqrt{(-7)^2+(-2)^2}=2\sqrt{49 + 4}=2\sqrt{53}$.
Length of $BC$: $d_{BC}=2\sqrt{(5 - 7)^2+(5-(-3))^2}=2\sqrt{(-2)^2+8^2}=2\sqrt{4 + 64}=2\sqrt{68}$.
Length of $AC$: $d_{AC}=2\sqrt{(-2 - 7)^2+(3-(-3))^2}=2\sqrt{(-9)^2+6^2}=2\sqrt{81+36}=2\sqrt{117}$.

Step3: Calculate total length of border

$L=d_{AB}+d_{BC}+d_{AC}=2\sqrt{53}+2\sqrt{68}+2\sqrt{117}\approx2\times7.28+2\times8.25+2\times10.82=14.56 + 16.5+21.64=52.7$.

Step4: Calculate total cost

The cost per foot is $\$1.50$. So the total cost $C = 1.5\times L=1.5\times52.7 = 79.05\approx79$. So the answer for part B is C.

Answer:

A. A. $2\sqrt{(5 - 7)^2+(5-(-3))^2}$
B. C. $\$79$